Page 136 - Process Modelling and Simulation With Finite Element Methods

Multiphysics 123

3.3 Unsteady Response of a Nonlinear Tubular Reactor

Ramirez [l] [6] reports a simulation of the adiabatic tubular reactor where heat
generation effects are appreciable. Generally, tubular reactor design estimates
follow from steady-state 1-D ODE simulations. In the model of Ramirez, the
reactor starts up cold or is subjected to perturbations of its steady operation
which convect through the system before returning to steady operation. In this
regard, such transient effects are important considerations for the safe, stable and
controlled operation of tubular reactors.
Ramirez treats first order chemical reaction with heat generation. Thus only
the mass transport equation for one species and energy transport equation, coupled
through the temperature dependence of the reaction flux and the heat generation by
reactive flux, need be considered. Interestingly, Ramirez solved the highly
coupled, nonlinear equations by a technique of quasilinearization with finite
difference techniques. The solution at the current time and the linearization of the
equations about that solution are used to predict the profiles of concentration and
temperature at the next time step. The procedure is iterated until convergence at
the new time step is achieved. The prediction and correction steps involve solution
of sparse linear systems. This is, of course, the same procedure as used by
FEMLAB, except it is the finite element approximation and the associated sparse
linear system that is solved iteratively by Newton’s method.
Governing equations are given here in dimensionless form:
a0 a a20 ao +
-
r2 - B,l-exp (-QQ / 0)
at 3 ax2 ax
ar a2r ar (3.7)
-
r2 - B2rexp (-QQ / 0)
at ax2 ax -
subject to boundary conditions on the reactor inlet and outlet:

(3.8)

The former are called Danckwerts boundary conditions [7]. The initial
conditions for temperature are uniform everywhere at @=I. Ramirez [l]
considers two different liquid phase reactions. The first is a reactor with an
intermediate conversion at a single steady state. The second is a triple steady
state. The Peclet numbers for heat and mass transfer are taken as identical for